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Dominion of some graphs

Julian Allagan, Benkam Bobga

TL;DR

This work defines the dominion $\zeta(G)$ as the number of minimum dominating sets and studies its behavior across standard graph constructions. It delivers explicit, modulo-$3$ based formulas for $\zeta(P_n)$, a $2^n$ closed form for sun graphs, and a conjectured closed form for cycles $C_n$, alongside analysis of cycles derived from paths. It then analyzes the join operation $G_1\vee G_2$, providing gamma-value regimes and several bounds and exact counts for $\zeta(G_1\vee G_2)$, including special cases like complete multipartite graphs. Overall, the results deepen understanding of how the multiplicity of optimal domination patterns evolves under graph composition, with potential implications for network domination analysis.

Abstract

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a minimum dominating set. The dominion of a graph G, denoted by zeta(G), is the number of its gamma-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs.

Dominion of some graphs

TL;DR

This work defines the dominion as the number of minimum dominating sets and studies its behavior across standard graph constructions. It delivers explicit, modulo- based formulas for , a closed form for sun graphs, and a conjectured closed form for cycles , alongside analysis of cycles derived from paths. It then analyzes the join operation , providing gamma-value regimes and several bounds and exact counts for , including special cases like complete multipartite graphs. Overall, the results deepen understanding of how the multiplicity of optimal domination patterns evolves under graph composition, with potential implications for network domination analysis.

Abstract

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a minimum dominating set. The dominion of a graph G, denoted by zeta(G), is the number of its gamma-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs.
Paper Structure (3 sections, 9 theorems, 4 equations, 1 figure)

This paper contains 3 sections, 9 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dominion of Some Graphs
  3. Dominion of Join of Graphs

Key Result

Proposition 1.2

If $G$ is a sun graph on $2n\ge 6$ vertices, then $\zeta(G)=\sum_{i=0}^n{n\choose i}=2^n.$

Figures (1)

  • Figure 1: A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$

Theorems & Definitions (31)

  • Example 1.1
  • Proposition 1.2
  • proof
  • Claim 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • Claim 2.4
  • proof
  • ...and 21 more